Abstract: - A mathematical model for ignition and turbulent combustion of polydispersed dust-airmixtures in cylindrical vessel is developed. It uses both deterministic methods of continuum mechanics ofmultiphase flows to determine the mean values of the gaseous phase parameters, and stochastic methods todepict the evolution of polydispersed particles in it and fluctuations of paremeters. The equations ofmotions for particles consider the influence of random turbulence pulsations in gas flow. Thermaldestruction of dust particles, vent of volatiles, chemical reactions in the gas phase, carbon skeletonheterogeneous oxidation by , and chemical reaction with are the processes essential fordescribing dust and particulate phases. The mathematical model developed makes it possible to investigatethe peculiarities of polydispersed organic dusts ignition and combustion and the influence of flownonuniformities on the ignition limits.2O2CO2H O

Investigation of the ignition and combustion ofpolydispersed mixtures is important for thedescription of dust explosion accidents, developingpreventive measures in industry, and working outmethods for testing ignitability and explosibility ofdusts.[1-4] To describe the processes taking place inmotor chambers and burners of different types and todevelop measures for improving their perfomancecharacteristics the models for ignition andcombustion of polydispersed mixtures are created.The processes of polydispersed mixturescombustion are rather complex due to the effects ofturbulent interaction of gaseous and dispersedphases, heat and mass transfer, thermal destructionand volatiles extraction, gaseous and heterogeneouschemistry [5-11]. All of the processes are greatlyinfluenced by turbulence [12-17]; and apolydispersed phase which is distributedstochastically makes the experimental investigationsvery difficult because of nonreproducibility of theinitial conditions at microscales and the lack ofpossibility to vary a single model parameter keepingthe rest of them constant in the experiments. Theignition limits sensitivity to the properties of mixture– composition, size and shape of particles,turbulence, dust concentration and distribution, gas-phase composition - can hardly be investigatedexperimentally as in this case only groups ofparameters vary. Besides, the stabilization of themixture initial properties in different experiments ishardly possible. Hence, numerical investigationsprovide a unique possibility to study the influenceof dust parameters on the ignition process.The present research aims at obtainingnumerical solutions for turbulent combustion in theheterogeneous mixtures of gas with polidispersedparticles, and studying theoretically the influence ofgoverning parameters on the mixturecharacteristics. It uses both deterministic methodsof continuum mechanics of multiphase flows (todetermine the mean values of the gaseous phaseparameters) and stochastic methods (to depict theevolution of polydispersed particles in it andfluctuations of parameters). Thermal destruction ofdust particles, vent of volatiles, chemical reactionsin the gas phase, and carbon heterogeneousoxidation by and and chemical reactionwith are the processes essential for themodels of phase transitions and chemical reactions[18]. Thus accounting for these effects wasincorporated into the current numerical model.2O2CO2H O

2 Mathematical model

The characteristic peculiarities of themathematical model are as follows:• The Euler approach is used to describe the gasphase, and the Lagrange method enables tomodel that of dust.• The turbulent flows take place into both gasand particulate phases. To simulate the gas6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)Rhodes, Greece, August 20-22, 2008ISSN: 1790-509549ISBN: 978-960-6766-97-8

phase flow we use the modified k-epsilon modeland investigate combustion and heat and masstransfer in the phase [19-20].• Combustion is developing both on the particlesurface and in the gas phase. Three reactions areassumed to take place on a particle surface.Futhermore, there is a volatiles (L) extractionfrom the carbon dust.( ),pL →L

,222COOC →+22C CO CO+ →

2 2C H O CO H+→ +

Here L denotes generalized volatile component.It is supposed that the volatiles being extracted fromparticles consist of the following components:L O CO CO HO N CH HO CO CO HO N CH H= + + + + + +ν ν ν ν ν ν ν2 2 2 2 422020 020202040We use the modified k-epsilon model to describe thegas phase behavior. The generalization of this modelconcentrates on the influence of other phases (mass,momentum, energy fluxes from the particles phase)as well as the combustion and heat and mass transferin the gas phase. The system of equations for the gasphase was obtained by Favre averaging the systemof multicomponent multiphase media withweightαρ. [8, 20].To simulate the polydispersed particulate phasea stochastic approach is perfomed, for that a groupof representative particles is distinguished. Theeffect of the gas mean stream and pulsations ofparameters in the gas phase on the motion ofparticles is considered. Gas flow properties (themean kinetic energy and the rate of pulsationsdecay) make it possible to model the particlesstochastic motion determining the pulsations changefrequency under assumptions of the Poisson flow ofevents.We model a great amount of real particles byassembling model particles (their number is of theorder of thousands). Each model particle ischaracterized by a vector of values, representing itslocation, velocity, mass and other properties. Thefollowing vector is determined for each modelparticle:{},,,,,,,ssiN m m r u w Tω  ,i N.1,...,p=When a particle is burnt out, its massm0i=, andthis particle is excluded from calculations. Theprocesses of adhesion and fragmentation are notconsidered.The parameter is used to model dust phaseturbulent pulsations. It represents the pulsationvelocity vector added to the gas velocity vectoriwv,thus a stochastic force resulting from theinteraction with turbulized gas is calculatedjointly with a resistance force. The laws ofparticle motion are as follows:( )( )ifi si ti fi si tiiidum m m m m m g fdtdrudtri++ = + + +=

where the force affecting the particle consists ofgravity, Archimedean forces, resistance force and astochastic force resulting from the interaction withturbulized gas. We neglect several forces such asassociate mass forces, Magnus force, Basse force.Fig. 1. Scheme of particle burning.It is supposed that the volatiles are extractedfrom dust (Fig. 1), and three brutto reactions areassumed to take place on a particle surface:( ),pL L→

Introducing new variables and modifying thediffusion coefficient we obtain the solution forthe complicated problem with the finite boundaryconditions. Second, one more correction is appliedto the diffusion coefficient to take underD6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)Rhodes, Greece, August 20-22, 2008ISSN: 1790-509550ISBN: 978-960-6766-97-8

(λdenotes a thermal conductivity calculated whilethe particle temperature equalssiT; is thetemperature of a surrounding gas).TThe specific internal energy of particlesieis equalto the ratio of full energysiEto the particle mass:0fi L fisi si s ti Tsii i i i im c mE m c m cem m m m m⎛ ⎞= = + + +⎜ ⎟⎝ ⎠si LT h.The term Qsirepresents heat extraction due tochemical reactions on theith model particle surface.The term determines the energy influx to theth model particle from the energy source.eieiFurther the fluxes from model particles areconsidered, and their recalculation into gas-phaseequations is performed.The boundary conditions for the gas phase areconstructed in accordance with the followingconsiderations: the walls of the cylindrical domainare noncatalytic, the gas velocity is zero at the walls,and the averaged gas motion has cylindricalsymmetry. This leads to Neumann’s conditions fortemperature and mass fractions at the vessel walls:0, 0, 0kx rY Tv vn∂ ∂= = =∂n=∂,where n is the normal vector to the wall. Theboundary conditions for turbulent parameters andkεare constructed according to the Lam-Bremhorstmodel:0=k,0nε∂=∂.Boundary conditions at the symmetry axis( 0)r=

are constructed to satisfy those of the flowcharacteristics continuous differentiability:0, 0, 0, 0, 0, 0k xrY v T kvr r r r rε∂∂ ∂ ∂ ∂== = = = =∂ ∂ ∂ ∂ ∂.The following boundary conditions forparticulate-phase motion are considered: particlesare bouncing against the solid walls of thecylindrical vessel without kinetic energy loss. Itmeans that a particle velocity normal componentchanges its sign after the particle collides with thewall, and the tangential component keeps its valueand direction.The initial conditions for the gas phase include:zero average velocity, given gas temperaturedistribution, the species mass fraction, and theturbulence parameters (taken uniform in ourinvestigations).As to initial conditions for the particulatephase, each model particle is distributed within thecylindrical computational domain using a givendensity of the particulate phase. The exact modelparticle localization is generated by randomcounters. The particle temperature, shape, andcondensed volatiles quantity are constant, thus wedetermine mass and other model particle properties.Ignition process was modelled as an energyrelease in a relatively small volume inside thevessel with the power asa given function of time.Calculations areperformed in terms ofcylindrical geometrywith the uniform grid61x41 and 10000 modelparticles. Each time stepcontains the modelparticles motion calculations, determining fluxesfrom particles to the gas phase and recalculatingthem to the grid. Particle motion was computedusing an iterative implicit algorithm for eachparticle independently. Then two iterations wereundertaken to calculate gas dynamics parametersconcerning fluxes from the particulate phase. Ateach iteration the space splitting in x and rcoordinates is performed as well as the splitting inthe following physical processes: chemistry, sourceterms and turbulent energy production (local part ofthe equations), convection (hyperbolic part),diffusion (parabolic part). The applied operatorsplitting techniques represent the general operator()Lt∆transferring the parameters vectornPto6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)Rhodes, Greece, August 20-22, 2008ISSN: 1790-509551ISBN: 978-960-6766-97-8

the next time step: .1nP+( )1n nP L t P+= ∆ ()L t∆

is then represented in a special form by means oftwo operators, each split into three parts: parabolic,hyperbolic and local. The local part is solvedimplicitly using an iterative algorithm independentlyfor each grid node. The hyperbolic part was solvedusing explicit flux-corrected transport (FCT)techniques; the parabolic part was solved implicitlyusing 3-diagonal matrix solvers for linear equations.The scheme gas dynamic part was validated bycomparing with standard solutions. The physicalmodel for gas-particles flows and phenomenologicallaws for phases interactions were validated bycomparing the numerical modeling results ofmultiphase hydrocarbon-air mixtures combustionwith the shock-tube experiment results. The dustcombustion model was compared with experimentson organic dusts combustion [8-11].

3 Results

The numerical simulations were performed forcarbon dust-air mixtures combustion in a closedcylindrical 1.25 m3vessel. The mixture was ignitedin the vessel center by energy release in a ball-shaped volume. The flame boundaries were detectedalong five central rays (in horizontal, verticaldirections and 45º to the horizon) as the points ofvolatiles oxidizing intensity maximal gradients oneach ray.According to the numerical computations thecombustion process can be divided into thefollowing stages: the initial flame ball formation at10-20 ms just after switching off an ignitor; thedeveloped flame stage at 20-50 ms; the final flamepropagation stage at 50-80 ms and further processesevolution. Time limits strongly depend on themixture composition, initial turbulization, initialdust-phase density, oxygen concentration, etc.Figures 2, 3, 4, 5 illustrate combustion parametersdynamics in time: temperatureT, dust-phase mass( )PMt, mean pressure in the vessel()P tandflame front position( )fRtrespectively.( )P tand( )PMtgraphs correlate greatly:pressure increases with dust-phase mass diminutiondue to its converting into the gas phase. Thepressure maximum is observed at 70-80 ms afterflame front passing. It is explained by the fact thatalmost all dust phase is heated at the moment, andpyrolysis amounts to its maximum. As timeexceeds 80 ms, the pressure growth stops abruptly,and dust-phase mass approximates to the value oftar. The effect is caused by the completion ofvolatiles pyrolysis and carbon skeleton oxidation.

Considering the front position graph (figure 5),blue and red lines (being in close agreement)correspond with the flame front position accordingto the gradient maximum of oxidizer andtemperature. Discrepancy is only essential at theignition stage as a result of energy source action.By 65 ms fore front approaches to the vessel wall.Up to 57 ms it is moving behind the flame centralaxis, and front width is nearly constant. Then thefront return to the vessel center is registered, andafter 65 ms its motion is unstable, because as theflame fore front comes to the walls, the flamewidth and intensity increase rapidly, whileintensity peak decreases. As a result theposition=1/10 of intensity maximum moves back.It is derived from the graphs that at thebeginning the velocity of flame propagationequals 9,6 m/s, and as the flame approaches thewalls it equals 5,8 m/s. The mean velocity valueis 7,9 m/s.The set of numerical simulations wasperformed for different initial density values:ρ= {0,1; 0,4;} kg/m3. Figures 6-7 illustrate theevolution of the mean pressure and total( )P tdust-phase mass( )fMtwith the time for thepointed densities correspondingly.

Fig. 6. Pressure and dust phase mass evolution intime for0.1ρ=.

Fig. 7. Pressure and dust phase mass evolution intime for0.4ρ=.All experiments testify a strong correlationbetween the pressure and the particles mass;pressure increases first of all due to gase-phasemass augmentation (influencing gas-phasetemperature). For the case of not very high initialdensity (0.1-0.2 kg/m3) the pressure increases upto 12 bar, for higher density – up to 9-11 bar.When0.4ρ=kg/m3, the pressure abruptdecrease is observed after its maximum value.This is a result of irreversible reactions22C CO CO+→2 2C H O H CO+ → +,becoming more intensive due to fuel surplusand the lack of oxidizer. Experiments providethe same effect mainly because of heat fluxesthrough walls.4 Conclusions• A computational model describing the turbulentcombustion dynamics in heterogeneousmixtures of gas with polydispersed particles ismodified. The present model takes into accountcomplicated kinetics thus specifying solutions.The model enables one to determinepeculiarities of turbulent combustion ofpolydispersed mixtures.• The integral combustion parameters dynamicswas analyzed for the case of air-dust mixture.• The initial dust-phase density effect on pressureand dust-phase mass were invstigated. It isshown that after pressure equals its maximum6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)Rhodes, Greece, August 20-22, 2008ISSN: 1790-509553ISBN: 978-960-6766-97-8